K12MATH006: Math Grade 6

Unit 3: Expressions and Equations

Have you ever noticed that patterns are found everywhere? We use
expressions and equations to put those patterns into mathematical
sentences. Comparing the equations allows people to recognize how one
pattern compares to another. For example, if you have a cell phone, you
or a parent might have done some “comparison shopping” before signing up
for the plan. Without even knowing it, you used an equation to compare.
If you and your friends want to rent bicycles while on vacation, you
might use an equation to figure out how much it would cost for eight
friends to rent bikes. You’d use the same equation if only five friends
plan to rent. Equations and expressions involve variables, which helps
open lots of doors into algebra. While working through this unit, you
will be constructing the basic building blocks of algebra, which is
important to your mathematical future.

Unit 3 Time Advisory
Completing this unit should take you approximately 13 hours.

☐ Subunit 3.1: 15 minutes

☐ Subunit 3.2: 2 hours and 30 minutes

☐ Subunit 3.3: 3 hours and 15 minutes

☐ Subunit 3.4: 4 hours and 15 minutes

☐ Subunit 3.5: 2 hours and 45 minutes

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to:

When students first learn about multiplication, teachers often relate
it to addition. Perhaps you have heard the phrase “multiplication is
repeated addition.” An exponent is similar because an exponent helps
mathematicians represent repeated multiplication. This subunit will
teach you about how to solve for and write exponents.

Instructions: Watch the video about exponential notation. Take
notes on the 2nd slide (0:20). There is important vocabulary in red
text that you will want to take notes on. As you watch, pay
attention to how the instructor uses those vocabulary words
throughout the video. The slide starting at 1:19 is also very
valuable; notice that multiple ways to verbally say x2 and x3 exist.
You want to be familiar with both “x squared” and “x to the second
power.”

Starting at 3:13, the instructor begins evaluating some exponents.
At this time, you only need to be able to do and understand #1.
Feel free to watch him solve the other three examples; do not worry
if the negative numbers or the multiplying with a variable are
confusing. You will learn those concepts throughout your math
career.

Watching this video and taking notes should take you approximately
15 minutes.

Mathematicians have a set of rules called order of operations; the
rules help everyone know what steps to take when solving a set of
numbers. As a mathematician you need to know these rules and how to
apply them because you will find that you need them for both simple and
complex problems you will encounter.

Instructions: Read about Keisha and counting birds. Take notes about
the difference between an equation and an expression. You will
continue to use those two words throughout this unit. Take notes
about order of operations. This will be an important concept to
remember whenever you are solving number sentences.

It’s often helpful to work the problems step-by-step to help
organize your information. A benefit to organizing your work
step-by-step is if you make a mistake, it’s easy to go back and
figure out where the mistake was made. If you try to do mental math,
you might end up with an incorrect answer. If you have not shown all
your work, finding your error will be hard.

Look at the example problem below. The student first wrote out the
entire problem, then solved the exponent operation and re-wrote the
entire problem with just the 42 solved. The next step was to see
that multiplication comes next, so the third line shows everything
written out with the 5 x 2 solved. Now the student is ready to
add/subtract from left to right. The fourth line shows the first
addition part solved with the remaining number sentence staying the
same. The final two lines show the last two operations solved as
they show up left to right. The number sentence gets shorter with
each line because the student solves one operation and then
re-writes everything else just as it was given in the problem.

3 + 42 – 5 x 2 + 9

3 + 16 – 5 x 2 + 9

3 + 16 – 10 + 9

19 – 10 + 9

9 + 9

18

Continue doing the examples. In section III, you will get to be a
math detective. Take time to check each of Joaquin’s problems and
make sure he followed the proper steps. Do all example problems.
Write down any vocabulary words that you have not already added to
your notebook.

Watch the videos. Notice how the instructors show each of their
steps as they work through the problem. Try to solve the expression
before the instructor so you can check your work. Pause the video
while you are working.

Work the “Time to Practice” problems for additional review.

Taking notes, watching the videos, and working through the examples
should take you approximately 1 hour.

Instructions: Do “Practice Set A,” “Practice Set B,” and “Practice
Set C” (exercise 19 in this last set is a challenge question). Check
your solution with the “Show Solution” button next to each problem.
Remember to show each step of your work. If you get a problem
incorrect, go back through each of your steps and find the error.

Remember in subunit 3.2 when you learned about the different between an
equation and an expression? Now you are going to focus on expressions
and how to evaluate expressions that involve variables (a variable is a
letter representing a number). Expressions can represent situations that
happen in your daily life. For example, as you will see in the first
resource, an expression can be used to figure out a weekly paycheck
based on the number of hours someone works.

Instructions: Read about Lydia and Bart working part time jobs.
Think about the steps that would go into solving this initial
question. Continue reading and taking notes under the “Guidance”
section. Solve the examples step-by-step on paper (not just
mentally); the process of substituting numbers for variables and
remembering to follow the order of operations shouldn’t be something
that is rushed. Continue doing all the guided practice, watch the
short video, and complete the practice problems.
Reading, taking notes, and practicing this concept should take you
approximately 30 minutes.
Standards Addressed *(Common Core)*:
- [CCSS.Math.Content.6.EE.A.1](http://www.corestandards.org/Math/Content/6/EE/A/1)
- [CCSS.Math.Content.6.EE.A.2](http://www.corestandards.org/Math/Content/6/EE/A/2)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial 3.0 Unported
License](http://creativecommons.org/licenses/by-nc/3.0/). It is
attributed to CK-12.

3.3.2 Writing Expressions for a Given Situation
- Explanation: CK-12: “Patterns and Expressions”

Link: CK-12: [“Patterns and
Expressions”](http://www.ck12.org/algebra/Patterns-and-Expressions/lesson/Patterns-and-Expressions/) (HTML)
Instructions: Read about Jeremy wanting to covert a phrase into
algebra. Continue reading about algebraic expressions and completing
the examples. Watch the first video about patterns.
Extra Challenge: The second video takes these concepts to a
higher-than-sixth-grade level. Feel free to watch it to “see where
you are going” in math. However, don’t get overwhelmed or
intimidated. The video uses equations instead of expressions. The
rules and ideas are the same, but since the instructor is solving
for a specific day, we use an equal sign.
Reading and watching the first video take should take you
approximately 15 minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.A.2a](http://www.corestandards.org/Math/Content/6/EE/A/2/a)
- [CCSS.Math.Content.6.EE.A.2b](http://www.corestandards.org/Math/Content/6/EE/A/2/b)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial 3.0 Unported
License](http://creativecommons.org/licenses/by-nc/3.0/). It is
attributed to CK-12.

Did I Get This? Activity: College of the Redwoods Mathematics:
Prealgebra Textbook: “Chapter 3, Section 1: Mathematical
Expressions - Exercises”

Link: CK-12: [“Single Variable
Expressions”](http://www.ck12.org/algebra/Single-Variable-Expressions/lesson/Single-Variable-Expressions/) (HTML)
Instructions: Read about Joshua trying to figure out his summer pay.
Notice in the introduction there is the expression “20x.” Continue
reading and taking notes about rules when working with variables -
specifically how to write variables and apply them to multiplication
and division problems. Work through the examples and guided
practice. Write down the vocabulary words that are not already in
your notebook.
Skip the first video. Watch the second and third video. Pause the
videos and solve each of the expressions before the instructor.
Remember to follow the order of operations and show each of your
steps.
Work through the practice problems.
Taking notes, watching the videos, and completing the practice
problems should take you approximately 30 minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.A.1](http://www.corestandards.org/Math/Content/6/EE/A/1)
- [CCSS.Math.Content.6.EE.A.2b](http://www.corestandards.org/Math/Content/6/EE/A/2/b)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial 3.0 Unported
License](http://creativecommons.org/licenses/by-nc/3.0/). It is
attributed to CK-12.

Link: CK-12: [“Expressions for Real-Life
Situations”](http://www.ck12.org/algebra/Expressions-for-Real-Life-Situations/lesson/Expressions-for-Real-Life-Situations/) (HTML)
Instructions: Read through the introduction. Pay close attention as
you read the first paragraph under the “Guidance” section. Take each
problem slowly; think about what type of operation would be used to
write an expression for each situation. Continue working on the
examples and guided practice. Skip the video as it covers
higher-level expressions with negative integers. Complete the ODD
practice exercises.
Taking notes and completing the practice problems should take you
approximately 30 minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.A.1](http://www.corestandards.org/Math/Content/6/EE/A/1)
- [CCSS.Math.Content.6.EE.A.2a](http://www.corestandards.org/Math/Content/6/EE/A/2/a)
- [CCSS.Math.Content.6.EE.A.2b](http://www.corestandards.org/Math/Content/6/EE/A/2/b)
- [CCSS.Math.Content.6.EE.A.2c](http://www.corestandards.org/Math/Content/6/EE/A/2/c)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial 3.0 Unported
License](http://creativecommons.org/licenses/by-nc/3.0/). It is
attributed to CK-12.

Did I Get This? Activity: Howard County Public School System’s
“Word Problems”

Instructions: This activity has eight problems. Two problems, #5
and #8, involve inequalities, which you will learn later in this
unit. Consider how you might solve them, but don’t get too caught up
in the exact expression. Complete all other word problems.

When you are finished check your answers
here (PDF).
If your variable is a different letter than the one on the answer
key, that is fine. Sometimes mathematicians always use the same
variable, like x; other times mathematicians use a variable that
best suits the situation.

Link: YouTube: Mathispower4u: James Sousa’s [“Introduction to the
Distributive
Property”](http://www.youtube.com/watch?v=6wj-H3SZQbQ) (YouTube)
Instructions: As you watch the video, write down the number
sentence. It also helps to draw the arrows (as seen in the video) to
guide you to multiply the outside number by each of the inside
numbers. Take notes while the instructor does each of the examples,
including drawing the rectangle to show how the distributive
property can be used to find area.
Taking notes and watching the video should take you approximately 15
minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.A.3](http://www.corestandards.org/Math/Content/6/EE/A/3)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution 3.0 Unported
License](http://creativecommons.org/licenses/by/3.0/). It is
attributed to James Sousa.

Instructions: From the table of contents, click on “The Distributive
Property” found beneath “3.3 Simplifying Algebraic Expressions.”
Read this section, take notes, and do the sample problems on pages
189 - 190. Understand that the number outside the parentheses needs
to be multiplied by each of the numbers inside the parentheses. As
you continue to work through higher-level algebra, you will extend
your knowledge of the distributive property.

Reading this section, taking notes, and working the sample problems
should take you approximately 15 minutes.

Solving an equation is like answering a question. Remember, an equation
is going to have an equal sign. Think of the equal sign as a balance
beam. Both sides are equivalent, and sometimes you have to work
backwards, or undo, one operation to figure out the unknown value (the
variable). In this subunit you will use substitution, like you did with
expressions - the concepts are the same.

Link: CK-12: [“Equations that Describe
Patterns”](http://www.ck12.org/algebra/Equations-that-Describe-Patterns/lesson/Equations-that-Describe-Patterns/) (HTML)
Instructions: Read the introduction and “Guidance” section. Take
notes on the bold words and information as you read. The question in
example A is a different type of question than in examples B and C.
You should feel comfortable being able to write equations as well as
using the substitution method to check if a specific value for a
variable results in a solution.
Listen to the first 30 seconds of the video. Pause it, and try to
write and solve the equation before continuing with the video. Watch
the remainder of the video so see if your equation and solving
method matched the solution. As the second example begins playing,
stop the video at the 2:15 mark. See if you can write and solve an
equation for the verbal expression before continuing the video.
Complete the exercises under the “Guided Practice” section, and
check your work with the given solutions. Stop there. Do not
continue with the next video or practice problems.
Taking notes, completing the practice problems, and watching the
video should take you approximately 30 minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.A.2](http://www.corestandards.org/Math/Content/6/EE/A/2)
- [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
- [CCSS.Math.Content.6.EE.B.6](http://www.corestandards.org/Math/Content/6/EE/B/6)
- [CCSS.Math.Content.6.EE.B.7](http://www.corestandards.org/Math/Content/6/EE/B/7)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial 3.0 Unported
License](http://creativecommons.org/licenses/by-nc/3.0/). It is
attributed to CK-12.

Instructions: Read through the amusement park problem and the
dilemma about the cost. Read through the “Guidance” section and
example problems. Take note of the key words that help hint toward
the operation or task. Complete the exercise under the “Guided
Practice” section.

Watch the first video. The instructor does a good job of really
explaining what 7x = 14 means. Take notes as you watch the video.
Listen at the 3:53 mark for the word “coefficient” - this is an
important vocabulary word to know. Write down the definition as you
hear it. Continue watching the video and take notes as you go.
Combining “like variables” is a skill the instructor shows you. This
concept will come up again as you continue studying algebra in the
future.

Complete the practice problems.

Taking notes, completing the practice problems, and watching the
videos should take you approximately 45 minutes.

Instructions: This page provides a series of practice problems that
you can answer and check online. Each question has a solution worked
out step-by-step if you need hints along the way. Practice the
one-step equations until you feel confident that you understand how
to solve for the unknown variable (recommended a minimum of 10
minutes of practice).

Practicing these one-step equations should take you approximately 15
minutes.

Instructions: Use the slider to help you understand the relationship
between inches and centimeters. Answer the question and check your
answer with the Submit Answer button. Do not move on until you have
the correct answer.

Completing this question and checking your work should take you
approximately 15 minutes.

Link: CK-12: [“Applications of One-Step
Equations”](http://www.ck12.org/algebra/Applications-of-One-Step-Equations/lesson/Applications-of-One-Step-Equations/) (HTML)
Instructions: As you read through the introduction, think about how
different the equations would look for the two situations about your
social network connection compared to your friends. Read through the
“Guidance” section and solve the problems in the “Examples” and
“Guided Practice” sections. Write out each equation and show the
steps to find the solution. Don’t just use mental math. As you move
into higher-level math, the examples will become more challenging,
but the steps will stay the same. Now is the time to really
understand how to write and solve equations.
Watch the video and solve each of the problems. The instructor goes
into helpful detail by using number lines, models, and number
sentences.
Solve questions 1 - 3 under the video for additional practice.
Reading this lesson, watching the video, and completing the practice
problems should take you approximately 45 minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.A.2](http://www.corestandards.org/Math/Content/6/EE/A/2)
- [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
- [CCSS.Math.Content.6.EE.B.6](http://www.corestandards.org/Math/Content/6/EE/B/6)
- [CCSS.Math.Content.6.EE.B.7](http://www.corestandards.org/Math/Content/6/EE/B/7)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial 3.0 Unported
License](http://creativecommons.org/licenses/by-nc/3.0/). It is
attributed to CK-12.

Instructions: Remember in the previous resource you learned about
independent and dependent variables. When you graph information from
a table to a graph the independent variable goes on the x-axis and
the dependent variable goes on the y-axis.

Complete the questions on the first page. Check your solutions and
read the commentary on pages 2 - 4.

Completing this checkpoint and checking your solution should take
you approximately 45 minutes.

Instructions: This checkpoint calls on you to apply your previous
learning about ratios to your current learning of equations. There
are three parts (a, b, and c) to this resource. Read the
introduction in the light gray box about the cars featured in the
magazine.

Part a: Fill in the blanks to complete the rate table. Check your
answer with the Submit Answer button. Do not move on until you have
the correct answer.

Part b: Click on “Part b” in the upper right corner. In case you are
not familiar with miles per gallon - a consumer wants to own a car
that has the highest number of miles per gallon. This allows the
person to travel a farther distance before needing to fill up the
gas tank (saves money!). Use the information in the gray box to
answer the question. Check your answer with the Submit Answer
button. Do not move on until you have the correct answer.

Part c: Click on “Part c” in the upper right corner. Use the
information in the gray box to answer the question. Check your
answer with the Submit Answer button. Do not move on until you have
the correct answer.

Completing the questions and checking your work should take you
approximately 30 minutes.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

3.5 Inequality

Have you ever solved a math problem where there was more than one
correct answer? Perhaps you were considering the minimum number of lawns
you needed to mow to buy a new skateboard. You might conclude that
mowing more than nine lawns would get you an adequate amount of money.
This means the correct answer for how many lawns to mow would be
anything larger than nine. This is where you would use an inequality. An
inequality is like an equation, but the equal sign is replaced with a
greater-than symbol or a less-than symbol. You probably have experience
with greater-than (>) and less-than symbols (<). There are often
infinite possible solutions for some inequalities (and sometimes there
are no solutions at all!). The rules of substitution and variables
remain generally the same.

Link: K12 Handhelds, Inc.:
*[Inequalities](http://k12opened.com/ebooks/math/ebook-inequalities/index.html)*:
“Overview” and “Graphing Inequalities” (HTML)
Instructions: This page will give you a really good summary of what
an inequality is and how to apply it. Read the “Overview” section;
take notes of each of the words in blue font. Read through the
examples, taking additional notes as you go. Do practice problems
1 - 11 (you can check answers 8 - 11).
In the “Graphing Inequalities” section, read and take notes.
Especially note the symbols and circles (shaded or not shaded) in
the outlined boxes. This is very important to graphing inequalities
on a number line. Read through the examples, and complete practice
problems 12 - 19.
Stop there for now. We will return to this page in the next subunit.
Reading this selection, taking notes, and completing the practice
problems should take you approximately 30 minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.B.8](http://www.corestandards.org/Math/Content/6/EE/B/8)
This resource is licensed under a [Creative Commons Attribution 3.0
United States
License](http://creativecommons.org/licenses/by/3.0/us/).

Instructions: Watch the video. Draw a number line and graph the
inequality on a number line. Notice how the instructor makes a
circle around the dash representing the number 4. This is because
the less-than sign represents the inequality.

Instructions: This page provides a series of practice problems that
you can answer and check online. Each question has a solution worked
out step-by-step if you need hints along the way. Practice graphing
inequalities and writing inequalities until you feel confident that
you understand how to graph and identify an inequality (recommended
a minimum of 10 minutes of practice).

Practicing graphing inequalities on a number line should take you
approximately 15 minutes.

Link: K12 Handhelds, Inc.:
*[Inequalities](http://k12opened.com/ebooks/math/ebook-inequalities/index.html)*:
“1-Step Addition and Subtraction Inequalities” (HTML)
Instructions: Scroll down to the section titled “1-Step Addition and
Subtraction Inequalities.” Read and take notes about how to solve
addition and subtraction inequalities. You will notice that solving
an inequality is similar to solving an equation. An important step
to remember is to always check your solution once you are complete.
This is a learning objective for sixth-grade students.
Reading and taking notes should take you approximately 15 minutes.
Standards Addressed (*Common Core*):
- [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
- [CCSS.Math.Content.6.EE.B.8](http://www.corestandards.org/Math/Content/6/EE/B/8)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution 3.0 Unported
License](http://creativecommons.org/licenses/by/3.0/).

Instructions: Watch the video and work the problems out with the
instructor. As a sixth-grade student, your learning objective is to
be able to use substitution to determine whether a given number will
make an inequality true.

Reflection: Write a paragraph comparing and contrasting the
difference between expressions, equations, and inequalities. How are
they the same? How are they different? Give some examples in your
own life where you might use one of these algebraic formats over
another.

Watching the video, taking notes, and writing the paragraph should
take you approximately 30 minutes

Link: Khan Academy’s
[“Inequalities”](https://www.khanacademy.org/math/algebra/linear_inequalities/inequalities/v/inequalities) (YouTube)
Instructions: Read the word problem at the start of the video. Stop
the video at the 0:30 mark and try to solve it on your own. Use the
instructors as a guide if you need help. Notice that two questions
are being asked. Use the inequality to help you find the solution
for how many tiles are needed to build the stone patio.
Watching this video and solving the problem should take you
approximately 15 minutes
Standards Addressed *(Common Core)*:
- [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
- [CCSS.Math.Content.6.EE.B.8](http://www.corestandards.org/Math/Content/6/EE/B/8)
Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-NoDerivs 3.0 Unported
License](http://creativecommons.org/licenses/by-nc-nd/3.0/). It is
attributed to the Khan Academy.

Instructions: Test your knowledge by completing this checkpoint.
Note: You must be logged into your Saylor Foundation School account
in order to access this checkpoint. If you do not yet have an
account, you will be able to create one, free of charge, after
clicking the link.